Question: The following system of equations is represented by the matrix equation $\text{A}\left[\begin{array} {ccc} x \\ y \\ z \end{array} \right]=\vec{b}$. $\begin{aligned}10x-5y&=20 \\15x+20y&=-8 \\12x+5y-20z&=30\end{aligned}$ ${A}=$ $\vec{b}=$ Represent each row and column in the order in which the variables and equations appear.
Solution: The Strategy A system of equations can be represented by a matrix equation $\text{A}\vec{x}=\vec{b}$, where $\text{A}$ is the coefficient matrix, $\vec{x}$ is the variables vector, and $\vec{b}$ is the constants vector. Each row of the matrix equation represents an equation in the system. [I need an explanation, please!] Representing the system of equations as a matrix equation We are given the system of equations: $\begin{aligned}10x-5y&=20 \\15x+20y&=-8 \\12x+5y-20z&=30\end{aligned}$ First, let's rewrite this system to show the coefficients of each variable. $\begin{aligned}{10}x+({-5})y+{0}z&=20 \\{15}x+{20}y+{0}z&=-8 \\{12}x+{5}y+({-20})z&=30\end{aligned}$ Now, the coefficient matrix can be written as follows. $\left[\begin{array} {ccc} {10} & {-5} & {0} \\ {15} & {20} & {0} \\ {12} & {5} & {-20} \end{array} \right]$ We can multiply this matrix by a column vector of variables and set it equal to a column vector with the values on the right side of the equations, as follows. $\left[\begin{array} {ccc} {10} & {-5} & {0} \\ {15} & {20} & {0} \\ {12} & {5} & {-20} \end{array} \right]\left[\begin{array} {ccc} x \\ y \\ z \end{array} \right] =\left[\begin{array} {ccc} 20 \\ -8 \\ 30 \end{array} \right]$ This is our matrix equation. Summary $\text{A}$ and $\vec{b}$ are shown below. $\text{A}=\left[\begin{array} {ccc} 10 & -5 & 0 \\ 15 & 20 & 0 \\ 12 & 5 & -20 \end{array} \right]~~~~~~~~~~~~ \vec{b}=\left[\begin{array} {ccc} 20 \\ -8 \\ 30 \end{array} \right]$